Are There Any Exceptions To Legally-Mandated Auto Insurance?

You pretty much have to have car insurance. There are some exception to this — but the list of legitimate exceptions is pretty short. You are not required to have auto insurance if:

• You live in New Hampshire. New Hampshire has no legal mandate for drivers to carry car insurance. It's still a good idea to carry insurance, as you will be required to pay should you be held responsible for an accident. But it is not legally required.
•  You own a car but you don't drive it. You don't need to buy liability coverage on a car that is in storage or waiting in your garage or backyard for repairs. It can be a good idea to invest in comprehensive coverage so that you're protected from various hazards that may befall your vehicle. But if you're not driving it, you're not required to carry liability.
•  You're driving, but it's not your car. If you borrow a friend or family member's car frequently, you'll want to get listed on their insurance. If you borrow or rent cars often, you may want to look into non-owner's insurance. This provides you with some protections, while not being as expensive as insuring a car of your own. By and large, you do not need insurance for one-time borrows, however.
•  You live in Virginia and are willing to pay the fine. In Virginia, you can opt-out of car insurance with a $500 annual fine. This provides you with no car insurance, but it may cost about the same as just buying the minimum amount of liability. It's an option, but a real viable one.

As you can see, the list is pretty short.

Generally speaking, if you drive, you need auto insurance. There are a few highly-circumstantial exceptions, but no real way around this requirement in the long run.